Integrable Representations of Involutive Algebras and Ore Localization

Abstract

Let A be a unital algebra equipped with an involution (·) † , and suppose that the multiplicative set S ⊆ A generated by the elements of the form 1 + a † a contains only regular elements and satisfies the Ore condition. We prove that:• Ultracyclic representations of A admit an integrable extension (acting on a possibly larger Hilbert space).• Integrable representations of A are in bijection with representations of the Ore localization AS −1 (which we prove to be an involutive algebra).This second result can be understood as a restricted converse to a theorem by Inoue asserting that representations of symmetric involutive algebras are integrable.2010 MSC: 16S (primary); 46L (secondary). * Brvargas@inst-mat.utalca.cl.